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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxpsd | Structured version Visualization version GIF version |
Description: Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brtxpsd.1 | ⊢ 𝐴 ∈ V |
brtxpsd.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brtxpsd | ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5154 | . . 3 ⊢ (𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V))) | |
2 | opex 5470 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | elrn 5900 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉) |
4 | brsymdif 5212 | . . . . . 6 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉)) | |
5 | brv 5478 | . . . . . . . . 9 ⊢ 𝑥V𝐴 | |
6 | vex 3466 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
7 | brtxpsd.1 | . . . . . . . . . 10 ⊢ 𝐴 ∈ V | |
8 | brtxpsd.2 | . . . . . . . . . 10 ⊢ 𝐵 ∈ V | |
9 | 6, 7, 8 | brtxp 35704 | . . . . . . . . 9 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ (𝑥V𝐴 ∧ 𝑥 E 𝐵)) |
10 | 5, 9 | mpbiran 707 | . . . . . . . 8 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 E 𝐵) |
11 | 8 | epeli 5588 | . . . . . . . 8 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵) |
12 | 10, 11 | bitri 274 | . . . . . . 7 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 ∈ 𝐵) |
13 | brv 5478 | . . . . . . . 8 ⊢ 𝑥V𝐵 | |
14 | 6, 7, 8 | brtxp 35704 | . . . . . . . 8 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ (𝑥𝑅𝐴 ∧ 𝑥V𝐵)) |
15 | 13, 14 | mpbiran2 708 | . . . . . . 7 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ 𝑥𝑅𝐴) |
16 | 12, 15 | bibi12i 338 | . . . . . 6 ⊢ ((𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
17 | 4, 16 | xchbinx 333 | . . . . 5 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
18 | 17 | exbii 1843 | . . . 4 ⊢ (∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
19 | 3, 18 | bitri 274 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
20 | exnal 1822 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) | |
21 | 1, 19, 20 | 3bitrri 297 | . 2 ⊢ (¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵) |
22 | 21 | con1bii 355 | 1 ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1532 ∃wex 1774 ∈ wcel 2099 Vcvv 3462 △ csymdif 4243 〈cop 4639 class class class wbr 5153 E cep 5585 ran crn 5683 ⊗ ctxp 35654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-symdif 4244 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-eprel 5586 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fo 6560 df-fv 6562 df-1st 8003 df-2nd 8004 df-txp 35678 |
This theorem is referenced by: brtxpsd2 35719 |
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